EXISTENCE THEOREM FOR LINEAR NEUTRAL IMPULSIVE DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

TitleEXISTENCE THEOREM FOR LINEAR NEUTRAL IMPULSIVE DIFFERENTIAL EQUATIONS OF THE SECOND ORDER
Publication TypeJournal Article
Year of Publication2018
AuthorsABASIEKWERE, UA, ESUABANA, IM, ISAAC, IO, LIPSCEY, Z
Secondary TitleLINEAR NEUTRAL IMPULSIVE EQUATIONS
Volume22
Issue2
Start Page135
Pagination12
Date Published01/2018
Type of Workscientific: mathematics
ISBN Number1083-2564
AMS34K11, 34K40, 34K45
Abstract

In this paper, we consider the second order linear neutral impulsive differential equation of the form

$$ \left\{\begin{array}{ll} \left[y\left(t\right)- p y\left(t-\tau \right)\right]^{{''} } + q\left(t\right)y\left(g\left(t\right)\right) = 0, &\quad t\ne t_{k},\\ \Delta \left[y\left(t_{k} \right)- p y\left(t_{k} -\tau \right)\right]^{{'} } + q_{k} y\left(g\left(t_{k} \right)\right) = 0, &\quad \forall t=t_{k} , \end{array}\right. $$

where $p\in R$, $ q_{k} \ge 0$, $ q \in P C \left(\left[t_{0} ,\infty \right), R_{+} \right)$, $ g\in C\left(\left[t_{0} ,\infty \right), R\right)$, $ \mathop{\lim }\limits_{t\to \infty } g \left(t\right)= \infty ,$ $\tau >0.$

We establish conditions for the existence of a positive solution with asymptotic decay by defining a map which satisfies the assumptions of Krasnoselkii’s fixed point theorem.
URLhttps://acadsol.eu/en/articles/22/2/1.pdf
DOI10.12732/caa.v22i2.1
Refereed DesignationRefereed
Full Text

REFERENCES

[1] D.D. Bainov and P.S. Simeonov, Oscillation Theory of Impulsive Differential Equations, International Publications Orlando, Florida, 1998.
[2] E.M. Bonotto, L.P. Gimenes and M. Federson, Oscillation for a secondorder neutral differential equation with impulses, Cadernos De Matematica, 09 (2008), 169-190.
[3] J. Cheng and Y. Chu, Oscillations of second-order neutral impulsive differential equations, Journal of Inequalities and Applications (2010) 1-29, doi: 10.1155/2010/493927.
[4] Andrey Antonov, Angel A. Dishliev, Angel B. Dishliev, Svetoslav Nenov, Mathematical Modeling of Discontinuous Processes, Scientific Research Publishing, Inc. USA, 2017, ISBN: 978-1-61896-439-7.
[5] L.H. Erbe, Q. Kong, and B.G. Zhang, Oscillation Theory for Functional Differential Equations, Dekker, New York, 1995.
[6] I. Gyori, and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Claredon Press, Oxford, 1991.
[7] Y.K. Hang, A.A. Nguraj and M.M. Arjunan, Existence results for impulsive neutral functional differential equations with infinite delay, Nonlinear Anal., 2 (2008), 209-218.
[8] E. Hern´andez, H. Henr´ıquez, M. Hern´an, and A. Mark, Existence results for abstract impulsive second-order neutral functional differential equations, Nonlinear Anal., 70, No. 7 (2009), 2736-2751.
[9] E. Hern´andez, C. Cuevas, and M. Rabello, The existence of solutions for impulsive neutral functional differential equations, Comput. Math. Appl., 58, No. 4 (2009), 744-757.
[10] I.O. Isaac, Z. Lipcsey, and U.J. Ibok, Nonoscillatory and oscillatory criteria for first order nonlinear neutral impulsive differential equations, Journal of Mathematics Research, 3, No. 2 (2011), 52-65.
[11] I.O. Isaac and Z. Lipcsey, Oscillations of scalar neutral impulsive differential equations of the first order with variable coefficients, Dynamic Systems and Applications, 19 (2010), 45-62.
[12] I.O. Isaac, Z. Lipcsey, and U.J. Ibok, Linearized oscillations in autonomous delay impulsive differential equations, British Journal of Mathematics and Computer Science, 4, No. 21 (2014), 3068-3076.
[13] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific Publishing Co. Pte. Ltd., Singapore, 1989.
[14] Y. Sun, Oscillation of certain second-order sub-half-linear neutral impulsive differential equations, Discrete Dynamics in Nature and Society (2011), 1-10, doi: 10.1155/2011/195619.
[15] A. Tripathy and S. Santra, Oscillation properties of a class of second order impulsive differential systems of neutral type, Functional Differential Equations, 23, No-s: 1-2 (2016), 57-71.
[16] Q. Wang and B.X. Dai, Existence of positive periodic solutions for a neutral population model with delays and impulse, Nonlinear Analysis A, 69, No. 11 (2008), 3919-3930.